The 5 Elements

We have dealt with all cases involving only pentagons, and with all those in which only one kind of figure is used: since, by Part 16, three hexagons do not form a solid angle.

These are the 5 bodies which the Pythagoreans, Plato, and Proclus called the world shapes. But I think it is not certain how they related these shapes to the bodies of the world.

Aristotle says that there were 5 such shapes which the philosophers related to the 5 elements:

1. Fire
2. Air
3. Earth
4. Water
5. Fifth Essence, or celestial matter

The characteristics of the shapes are compared with the properties of the simple bodies.

The cube stands upright on a square base expressing stability.

• This is characteristic of terrestrial matter, whose weight tends down to the lowest point
• It is commonly believed, that the whole Earth is at rest at the center of the World.

The octahedron, on the other hand, is viewed most appropriately suspended by opposite angles, as in a lathe, the square which lies exactly midway between these angles dividing the figure into 2 equal parts, just as a globe suspended by its poles is divided by a great circle.

This is an image of mobility, as air is the most mobile of the elements, in speed and direction.

The tetrahedron’s small number of faces is seen as signifying the dryness of fire, since dry things, by definition, keep within their own boundaries.

The large number of faces of the icosahedron, on the other hand, is seen as signifying the wetness of water, since wetness, by definition, is held within the boundaries of other things.

A few faces indicate that many from another body will be associated with them.

Furthermore, the plane trigon is proper to the tetrahedron, since the complete tetrahedral figure is a solid trigon, while the same trigon is not proper to the icosahedron, but, rather incidental to it, since the solid shape of the icosahedron is like a pentagon, not a trigon.

The tetrahedron’s point, rising from one face, is seen as expressive of the penetrating and dividing power of fire, while the blunt quinquelinear angle of the icosahedron expresses the filling power of humours, that is their power to wet.

The small thin tetrahedron shows the nature of fire; the large rounded mass of the icosahedron shows the nature of water, and as it were the shape of a drop.

The tetrahedron has a very large surface and a very small body; the icosahedron has a bodily mass much greater than its surface: just as in fire it is the form that predominates and in water it is the matter.

The dodecahedron is left for the celestial body, having the same number of faces as the Zodiac has signs. It can be shown that it has

1 1 1 the greatest volume of all the figures, just as the heavens enclose every­ thing else.

This analogy is acceptable, but not to Aristotle.

• He did not believe that the World had been created and thus could not recognize the power of these quantitative figures as archetypes, because without an architect there is no such power in them to make anything corporeal)
• Christians holds that the World:
  • had no previous existence,
  • was created by God in weight, measure, and number in accordance with ideas coeternal with Him

This analogy is acceptable, yet framed in this manner it has no force of necessity.

It admits of other interpretations, not only because certain properties are at variance within the analogy, but also because the dodecahedron and icosahedron correspond more closely with fire, and finally because the number of the elements and whether the Earth is at rest are matters much more open to dispute than is the number of the figures.

I do not blame Ramus, or Aristotle, for rejecting this disputed analogy.

24 years ago I found out a very different relation between these 5 shapes and the fabric of the world.

My introduction to Book 1 said that some of the ancients had been of this same opinion also, but had kept it secret, in the manner of their sect.

Copernican Astronomy and the Astronomy of the ancient Pythagorean Aristarchus of Samos describes the moving world as containing 6 paths or spheres surrounding the motionless Sun in the center

• The paths are separated from one another by large and unequal intervals.
• The outermost path is that of Saturn
• The next that of Jupiter
• Then that of Mars
• Then that of the Earth and the Moon
• Then that of Venus
• Lastly that of Mercury, the innermost.

These 5 shapes have a fundamental property that they can be drawn within a spherical surface so that their angles are on the surface, and can also be circumscribed about a spherical surface so as to touch it at the centers of their faces

Moreover, for any particular figure there is a particular interval between the 2 spheres defined in this way. Nothing seems more likely than that the 5 intervals between the six celestial spheres were taken by the Creator from the five figures, in an order such that the cube is to be imagined between the spheres of Saturn and Jupiter, the tetrahedron between those of Jupiter and Mars, the dodecahedron between those of Mars and the Earth, the icosahedron between those of the Earth and Venus, and the octahedron between those of Venus and Mercury.

This arrangement can be investigated numerically, and it has the force of necessity, not seeking anxiously for the number of the bodies but using the known number.

In addition, it is so well constructed that no one has attacked it in these twenty-two years, but even the pupils of Ramus, that hot-headed scholar, the scourge of Euclid, even they have been drawn to it, and it now excites so much interest that mathematicians are calling for a second edition to be brought out.^^

But it is not the purpose of this second [i.e. present] book to go into details of this theory. The reader will find more about it below, in Book 5, and something also in Book IV of the Epitome Astronomiae, where the true origin of these five solid figures is explained in metaphysical terms.

For their origin is not really from the properties of their solid angles but, rather, the properties of the solid angles are a consequence of the origin of the figures, being by their very nature something that comes later.